Quantum isotropy and the reduction of dynamics in Bianchi I

TL;DR

This paper develops a detailed embedding of the quantum isotropic model into the Bianchi I model within loop quantum gravity, establishing a precise equivalence of their dynamics and analyzing the residual diffeomorphism invariance.

Contribution

It provides a unique, covariant embedding of the isotropic loop quantum cosmology into the Bianchi I model, preserving key operators and dynamics, and clarifies the relationship between these models.

Findings

Established a unique embedding intertwining volume and Hubble rate operators.

Proved the embedding preserves the Hamiltonian constraints and dynamics.

Analyzed the residual diffeomorphism invariance in the embedding.

Abstract

The authors previously introduced a diffeomorphism-invariant definition of a homogeneous and isotropic sector of loop quantum gravity, along with a program to embed loop quantum cosmology into it. The present paper works out that program in detail for the simpler, but still physically non-trivial, case where the target of the embedding is the homogeneous, but not isotropic, Bianchi I model. The diffeomorphism-invariant conditions imposing homogeneity and isotropy in the full theory reduce to conditions imposing isotropy on an already homogeneous Bianchi I spacetime. The reduced conditions are invariant under the residual diffeomorphisms still allowed after gauge fixing the Bianchi I model. We show that there is a unique embedding of the quantum isotropic model into the homogeneous quantum Bianchi I model that (a) is covariant with respect to the actions of such residual diffeomorphisms, and (b) intertwines both the (signed) volume operator and at least one directional Hubble rate. That embedding also intertwines all other operators of interest in the respective loop quantum cosmological models, including their Hamiltonian constraints. It thus establishes a precise equivalence between dynamics in the isotropic sector of the Bianchi I model and the quantized isotropic model, and not just their kinematics. We also discuss the adjoint relationship between the embedding map defined here and a projection map previously defined by Ashtekar and Wilson-Ewing. Finally, we highlight certain features that simplify this reduced embedding problem, but which may not have direct analogues in the embedding of homogeneous and isotropic loop quantum cosmology into full loop quantum gravity.